Finite Group Databasebeta

← #11561

Subgroups — #11561

Order 256.

374 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 53 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2408C211yesmaximal, characteristic
2#1465C211yesmaximal, characteristic, complemented
3#1460C211yesmaximal, characteristic, complemented
4#1316C214yesmaximal, complemented
5C23×D4C411yescharacteristic
6C2×#121—21no—
7C2×#120C412yes—
8C2×C4×D4C2211yescharacteristic
9#431—21no—
10#407—21no—
11#394—21no—
12#392—21no—
13#390—21no—
14#380C411yescharacteristic
15#379C2212yes—
16#324—21no—
17#323—21no—
18#323C2212yes—
19#323C2212yes—
20C25C2×C412yes—
21C22×D4—22no—
22C22×D4—21no—
23C22×D4—21no—
24C22×D4—41no—
25C22×D4—41no—
26C22×D4—44no—
27C22×D4C2×C411yescharacteristic
28C23×C4C2×C411yescharacteristic
29C23×C4D411yescharacteristic
30C2×M16—41no—
31#121—44no—
32#120—44no—
33C4×D4—41no—
34C4×D4—41no—
35C2×C4⋊C4—21no—
36C2×C4⋊C4D411yescharacteristic
37C2×C22⋊C4—22no—
38C2×C22⋊C4—22no—
39C2×C22⋊C4—22no—
40C2×C22⋊C4C2×C412yes—
41C2×C22⋊C4D411yescharacteristic
42C2×C22⋊C4C2311yescharacteristic
43C2×C42D411yescharacteristic
44C4⋊C8—41no—
45C22⋊C8—22no—
46C22⋊C8—22no—
47C22⋊C8—41no—
48C4×C8—41no—
49#95—21no—
50C24—22no—
51C24—24no—
52C24—42no—
53C24—44no—
54C24—42no—
55C24—82no—
56C24C22⋊C411yescharacteristic
57C24C22×C411yescharacteristic
58C2×D4—24no—
59C2×D4—22no—
60C2×D4—22no—
61C2×D4—41no—
62C2×D4—42no—
63C2×D4—42no—
64C2×D4—44no—
65C2×D4—48no—
66C2×D4—44no—
67C2×D4—44no—
68C2×D4—44no—
69C2×D4—44no—
70C2×D4—41no—
71C2×D4—41no—
72C2×D4—82no—
73C22×C4—21no—
74C22×C4—21no—
75C22×C4—21no—
76C22×C4—21no—
77C22×C4—22no—
78C22×C4—42no—
79C22×C4—42no—
80C22×C4C22⋊C411yescharacteristic
81C22×C4C22⋊C411yescharacteristic
82C22×C4C22⋊C411yescharacteristic
83C22×C4C2×D411yescharacteristic
84C22×C4C2×D411yescharacteristic
85M16—82no—
86C2×C8—42no—
87C2×C8—42no—
88C4⋊C4—41no—
89C4⋊C4—42no—
90C22⋊C4—41no—
91C22⋊C4—41no—
92C22⋊C4—44no—
93C22⋊C4—44no—
94C22⋊C4—42no—
95C22⋊C4—82no—
96C42—21no—
97C42—21no—
98C23—21no—
99C23—21no—
100C23—24no—
101C23—21no—
102C23—21no—
103C23—21no—
104C23—41no—
105C23—41no—
106C23—41no—
107C23—41no—
108C23—42no—
109C23—42no—
110C23—42no—
111C23—44no—
112C23—44no—
113C23—42no—
114C23—42no—
115C23—42no—
116C23—84no—
117C23—88no—
118C23—88no—
119C23—84no—
120C23C2×C22⋊C411yescharacteristic
121C23#13112yes—
122D4—44no—
123D4—44no—
124D4—48no—
125D4—44no—
126D4—88no—
127C2×C4—21no—
128C2×C4—21no—
129C2×C4—22no—
130C2×C4—21no—
131C2×C4—21no—
132C2×C4—21no—
133C2×C4—42no—
134C2×C4—42no—
135C2×C4—42no—
136C2×C4—41no—
137C2×C4—41no—
138C2×C4—41no—
139C2×C4—42no—
140C2×C4—42no—
141C2×C4—42no—
142C2×C4—81no—
143C2×C4#9911yescharacteristic
144C2×C4#9911yescharacteristic
145C2×C4#10012yes—
146C8—82no—
147C8—82no—
148C22—21no—
149C22—21no—
150C22—22no—
151C22—42no—
152C22—42no—
153C22—42no—
154C22—44no—
155C22—44no—
156C22—41no—
157C22—41no—
158C22—41no—
159C22—44no—
160C22—41no—
161C22—41no—
162C22—81no—
163C22—82no—
164C22—82no—
165C22—88no—
166C22—88no—
167C22—82no—
168C22—82no—
169C22#40811yescharacteristic
170C22C2×#9911yescharacteristic
171C22C2×#10011yescharacteristic
172C4—21no—
173C4—22no—
174C4—41no—
175C4—41no—
176C4—41no—
177C4—82no—
178C4AΣL(1,𝔽16)12yes—
179C2—21no—
180C2—21no—
181C2—42no—
182C2—81no—
183C2—84no—
184C2#147111yescharacteristic
185C2C2×AΣL(1,𝔽16)11yescharacteristic
186C2#212211yescharacteristic
subgroupG/H
center, socleC22C2×#99
commutatorC24C22×C4
Frattini, cosocleC2×C22⋊C4C23
prime radical#2408C2
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #1465C2semidirect
3. #1460C2semidirect
4. #1316C2semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#24082
#14652
#14602
#13162